Periodic traveling waves in the ϕ4 model: Instability, stability, and localized structures

Meng-Meng Liu, Wen-Rong Sun, Lei Liu, P. G. Kevrekidis, and Lei Wang
Phys. Rev. E 107, 034210 – Published 23 March 2023

Abstract

We consider the instability and stability of periodic stationary solutions to the classical ϕ4 equation numerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figure eight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
6 More
  • Received 21 January 2023
  • Accepted 8 March 2023

DOI:https://doi.org/10.1103/PhysRevE.107.034210

©2023 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
  1. Physical Systems
Nonlinear Dynamics

Authors & Affiliations

Meng-Meng Liu and Wen-Rong Sun*

  • School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Lei Liu

  • College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

P. G. Kevrekidis

  • Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA

Lei Wang

  • School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

  • *sunwenrong@ustb.edu.cn

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 107, Iss. 3 — March 2023

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×